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defense-p3-ibid.tex
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\begin{frame}
\only<1-6>{\frametitle{Talk Outline}}
\only<7->{\frametitle{Lazily Evaluated Marginal Utility Roadmaps}}
\begin{tikzpicture}[font=\small]
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\tikzset{>=latex} % arrow heads
% START LEMUR
\node[fill=blue!5,draw=blue!10,rounded corners,minimum width=9.7cm,minimum height=5.8cm,anchor=north] (lemur) at (6,6.0) {};
\only<1-6>{\node[anchor=north west] at (lemur.north west) {Planner};}
\only<7->{\node[anchor=north west] at (lemur.north west) {LEMUR};}
% left side
\node[fill=blue!10,draw=blue!20,rounded corners,align=center,minimum height=1.5cm,minimum width=2.5cm,inner sep=0pt] at (3.3,4.5) {};
\node at (3.3,4.5) {\includegraphics[width=1.8cm]{build/roadmap-stack-short}};
\draw[->] (3.3,3.75) -- (3.3,3.0) node [pos=0.45,fill=blue!5,align=center,inner sep=2pt] {$G$};
% right side
\node[fill=blue!10,draw=blue!20,rounded corners,align=center,minimum height=1.5cm,inner sep=0pt] at (7.8,4.5) {
\qquad\qquad\; $ \arraycolsep=1.5pt \begin{array}{rcc}
w_{\ms{est}}(e) = & \lambda \, \hat{p}(e) \; + & (1\!-\!\lambda) \, \hat{x}(e) \\
w(e) = & & (1\!-\!\lambda) \, x(e)
\end{array} $
};
%\node[fill=black!3,draw=blue!20,inner sep=2pt] at (7,5.4) {\includegraphics[width=1.3cm]{build/pvx-utility-anytime-simple}};
\node[fill=black!5,draw=blue!20,inner sep=2pt] at (5.6,4.5) {\includegraphics[width=1.0cm]{build/pvx-linear-discounting-simple}};
\draw[->] (7.3,3.75) -- (7.3,3.0) node [pos=0.45,fill=blue!5,align=center,inner sep=2pt] {$w$};
\draw[->] (8.2,3.75) -- (8.2,3.0) node [pos=0.45,fill=blue!5,align=center,inner sep=2pt] {$w_{\ms{est}}$};
% START LAZYSP
\node[fill=blue!10,draw=blue!20,rounded corners,minimum width=7cm,minimum height=2.5cm,anchor=north] (lazysp) at (6,3.0) {};
\only<2->{\node[anchor=north west] at (lazysp.north west) {\strut LazySP};}
\only<3->{
\node[fill=blue!20,draw=blue!30,rounded corners,align=center,minimum height=1cm] (dynsp) at (4.3,1.7) {InnerSP};
}
\only<4->{
\draw[->] (dynsp.south) -- (4.3,0.8) -- (7.7,0.8) -- (7.7,1.2);
\node[fill=blue!10,align=center,inner sep=2pt] at (6,0.8)
{$\pi_{\ms{candidate}}$};
}
\only<5->{
\node[fill=blue!20,draw=blue!30,rounded corners,align=center,minimum height=1cm] (selector) at (7.7,1.7) {Edge Selector\\(e.g. Alternate)};
}
\only<6->{
\draw[->] (selector.north) -- (7.7,2.6) -- (4.3,2.6) -- (dynsp.north);
\node[fill=blue!10,align=center,inner sep=2pt] at (6,2.6)
{$E_{\ms{changed}}$};
}
% END LAZYSP
% END LEMUR
% top left side
\node[fill=blue!10,draw=blue!20,rounded corners,align=center,minimum height=1.5cm,minimum width=1.8cm,inner sep=0pt] at (3.3,7.0) {};
\node[fill=white,inner sep=0pt] at (3.3,7.0) {\includegraphics[width=1.4cm]{build/c-space-simple}};
\node[font=\scriptsize] at (2.95,6.7) {$\mathcal{C}_{\mbox{\tiny free}}$};
\draw[->] (3.3,6.25) -- (3.3,5.25) node [pos=0.55,fill=blue!5,align=center,inner sep=0pt] {\strut $\mathcal{C}$};
%\node[inner sep=4pt] (cspace) at (3.3,7.0) {$\mathcal{C}$-Space};
%\draw[->] (cspace) -- (3.3,5.25);
% top right side
\node[fill=blue!10,draw=blue!20,rounded corners,align=center,minimum height=1.5cm] at (7.9,7)
{$\arraycolsep=1.5pt \begin{array}{cl}
x(\xi)\!: & \mbox{execution cost} \;(\mathcal{C}_{\ms{free}}) \\
\hat{x}(\xi)\!: & \mbox{execution cost estimate} \\
\hat{p}(\xi)\!: & \mbox{planning cost estimate}
\end{array}$};
\draw[->] (7.3,6.25) -- (7.3,5.25) node [pos=0.55,fill=blue!5,align=center,inner sep=0pt] {\strut $x$};
\draw[->] (7.9,6.25) -- (7.9,5.25) node [pos=0.55,fill=blue!5,align=center,inner sep=0pt] {\strut $\hat{x}$};
\draw[->] (8.5,6.25) -- (8.5,5.25) node [pos=0.55,fill=blue!5,align=center,inner sep=0pt] {\strut $\hat{p}$};
% left side
\draw[->] (0.5,2.0) -- (2.5,2.0);
\node[fill=white,align=center,inner sep=2pt] at (0.5,2.0)
{$q_{\ms{start}}$\\$q_{\ms{dest}}$};
% right side
\draw[->] (9.5,2.0) -- (11.3,2.0);
\node[fill=white,align=center,inner sep=2pt] at (11.5,2.0)
{$\xi$};
\only<8->{
\draw[thick,rounded corners] (lazysp.south west) rectangle (lazysp.north east);
}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Lazy SP: The Inner Search}
\begin{tikzpicture}[font=\small]
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
%% graph start
%\coordinate (va) at ( 2.0,5.6);
%\coordinate (vb) at ( 3.5,6.3);
%\coordinate (vc) at ( 4.0,7.5);
%\coordinate (vd) at ( 5.5,6.1);
%\coordinate (ve) at ( 8.0,6.6);
%\coordinate (vf) at (10.0,5.6);
%\coordinate (vg) at ( 1.5,6.6);
%\coordinate (vh) at ( 4.0,5.1);
%\coordinate (vi) at ( 7.0,4.8);
%\coordinate (vj) at ( 6.5,7.1);
%\coordinate (vk) at ( 9.0,5.1);
%
%% start/goal highlighting
%\node[circle,fill=black!20,inner sep=0.1cm] at (va) {};
%\node[circle,fill=black!20,inner sep=0.1cm] at (vf) {};
%
%% candidate paths
%\draw[line width=0.2cm,color=black!30,line cap=round]
% (va) -- (vb) -- (vd) -- (vj) -- (ve) -- (vf);
%\draw[line width=0.2cm,color=blue!70,line cap=round]
% (vj) -- (ve);
%
%\draw[black!30] (va) -- (vb) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {5?};
%\draw[black!30] (vb) -- (vd) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {6?};
%\draw[black!30] (va) -- (vh) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {6?};
%\draw[black!30] (vh) -- (vd) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {7?};
%\draw[black!30] (vb) -- (vc) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {5?};
%\draw[black!30] (vc) -- (vj) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {8?};
%\draw[black!30] (va) -- (vg) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {3?};
%\draw[black!30] (vb) -- (vg) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {6?};
%\draw[black!30] (vb) -- (vh) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {5?};
%\draw[black!30] (vc) -- (vd) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {7?};
%\draw[black!30] (ve) -- (vf) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {7?};
%\draw[black!30] (vd) -- (vi) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {8?};
%\draw[black!30] (ve) -- (vi) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {8?};
%\draw[black!30] (vd) -- (vj) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {6?};
%\draw[black!30] (vf) -- (vk) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {3?};
%\draw[black!30] (vi) -- (vk) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {6?};
%\draw[black!30] (ve) -- (vk) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {6?};
%
%\draw[black,ultra thick] (vd) -- (ve) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {$\infty$};
%\draw[black!30] (ve) -- (vj) node[midway,circle,inner sep=0.02cm,fill=white,opacity=0.9,font=\scriptsize] {4?};
%
%\node[circle,fill=black,inner sep=0.05cm] at (va) {};
%\node[circle,fill=black,inner sep=0.05cm] at (vb) {};
%\node[circle,fill=black,inner sep=0.05cm] at (vc) {};
%\node[circle,fill=black,inner sep=0.05cm] at (vd) {};
%\node[circle,fill=black,inner sep=0.05cm] at (ve) {};
%\node[circle,fill=black,inner sep=0.05cm] at (vf) {};
%\node[circle,fill=black,inner sep=0.05cm] at (vg) {};
%\node[circle,fill=black,inner sep=0.05cm] at (vh) {};
%\node[circle,fill=black,inner sep=0.05cm] at (vi) {};
%\node[circle,fill=black,inner sep=0.05cm] at (vj) {};
%\node[circle,fill=black,inner sep=0.05cm] at (vk) {};
%
%\node[left=0.1cm of va] {$v_s$};
%\node[right=0.1cm of vf] {$v_t$};
% graph end
% here's an algorithm
\node[inner sep=0.2cm,fill=blue!10,rounded corners,anchor=north,minimum height=4.3cm,minimum width=7.5cm] at (6,7.75) {};
\only<2->{
\node[inner sep=0.2cm,fill=blue!30,anchor=north,minimum height=0.5cm,minimum width=7.5cm] at (6,6.15) {};
}
\node[inner sep=0.0cm,anchor=north] at (6,7.65)
{\hspace*{-0.6cm}\begin{minipage}{7.5cm}\small{
\algrenewcommand{\alglinenumber}[1]{}
\begin{algorithmic}[1]
\Function {\textsc{LazySP}}{$G, v_s, v_g, w, w_{\ms{est}}$}
\State $E_{\ms{eval}} \leftarrow \emptyset$
\State $w_{\ms{lazy}}(e) \leftarrow w_{\ms{est}}(e) \quad \forall e \in E$
\Loop
\vspace{0.05cm}
\only<1-2>{\State $p_{\ms{candidate}} \leftarrow \mbox{\sc InnerSP}(G, v_s, v_g, w_{\ms{lazy}})$}%
\only<3->{\State $p_{\ms{candidate}} \leftarrow \mbox{\sc DynamicSP}(G, v_s, v_g, w_{\ms{lazy}})$}%
\vspace{0.1cm}
%\vspace{0.2cm}
%\If {$p_{\ms{candidate}} \subseteq E_{\ms{eval}}$} \Comment If path is fully evaluated,
% \State \Return $p_{\ms{candidate}}$ \Comment return it
%\EndIf
\State \textbf{if} $p_{\ms{candidate}} \subseteq E_{\ms{eval}}$ \textbf{then return} $p_{\ms{candidate}}$
\vspace{0.1cm}
\State $E_{\ms{selected}} \leftarrow \mbox{\sc EdgeSelector}(G, p_{\ms{candidate}})$ %\Comment Select edges on path to process
\For {$e \in E_{\ms{selected}} \setminus E_{\ms{eval}}$} %\Comment For all unevaluated selected edges
\State $w_{\ms{lazy}}(e) \leftarrow w(e)$
\State $E_{\ms{eval}} \leftarrow E_{\ms{eval}} \cup e$ %\Comment Add to evaluated edge set
\EndFor
\EndLoop
\EndFunction
\end{algorithmic}
}\end{minipage}};
\only<4->{
\node[fill=blue!10,draw=blue!20,rounded corners,align=center,minimum height=1.8cm,inner sep=2pt,anchor=north] at (3.0,3.0) {
Motion Planning:\\
\vspace{-0.3cm}\\
$ \arraycolsep=1.5pt \def\arraystretch{1.5} \begin{array}{rl}
{\hat x}(e) = & ||e|| \\
x(e) = & \left\{ \arraycolsep=2pt \def\arraystretch{1.0} \begin{array}{cl}
||e|| & e \in C_{\ms{free}} \\
\infty & \mbox{otherwise}
\end{array} \right. \\
h_x(v) = & || v - v_g ||
\end{array} $
};
}
\only<5->{
\node[fill=blue!10,draw=blue!20,rounded corners,align=center,minimum height=1.8cm,inner sep=2pt,anchor=north] at (8.6,3.0) {
Maximizing Utility:\\
\vspace{-0.3cm}\\
\qquad\qquad\; $ \arraycolsep=1.5pt \def\arraystretch{1.2} \begin{array}{rcl}
w_{\ms{est}}(e) = & \lambda \, \hat{p}(e) \; + & (1\!-\!\lambda) \, \hat{x}(e) \\
w(e) = & & (1\!-\!\lambda) \, x(e) \\
h_w(v) = & & (1\!-\!\lambda) \, h_x(v)
\end{array} $
};
\node[fill=black!5,draw=blue!20,inner sep=2pt] at (6.35,1.85) {\includegraphics[width=1.0cm]{build/pvx-linear-discounting-simple}};
}
%Augment problem with inexpensive edge heuristic. (Show augmentation.)
%Each edge has binary state.
%Show outline of algorithm.
%Two core sub-routines: inner SP search, and edge selector.
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{LazySP: DynamicSP Algorithms}
\begin{tikzpicture}[font=\small]
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\node at (6,4) {\includegraphics{build/ibid-lazysp-plot,hideibid}};
%\node[draw=black!30,inner sep=0pt] at (3.5,5.0) {\includegraphics[width=2.5cm]{figs/herbarm-traj2-t031.png}};
\node[draw=black,inner sep=0pt,thick,anchor=north] at (5.15,7.325) {\includegraphics[width=2.0cm]{figs/herbarm-traj2-t031.png}};
\node[draw=black,fill=black!2,anchor=north] at (7.8,7.325) {\strut $h_w = (1 - \lambda) \, h_x$};
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Talk Outline}
\begin{tikzpicture}[font=\small]
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\tikzset{>=latex} % arrow heads
% START LEMUR
\node[fill=blue!5,draw=blue!10,rounded corners,minimum width=9.7cm,minimum height=5.8cm,anchor=north] (lemur) at (6,6.0) {};
\node[anchor=north west] at (lemur.north west) {LEMUR};
% left side
\node[fill=blue!10,draw=blue!20,rounded corners,align=center,minimum height=1.5cm,minimum width=2.5cm,inner sep=0pt] at (3.3,4.5) {};
\node at (3.3,4.5) {\includegraphics[width=1.8cm]{build/roadmap-stack-short}};
\draw[->] (3.3,3.75) -- (3.3,3.0) node [pos=0.45,fill=blue!5,align=center,inner sep=2pt] {$G$};
% right side
\node[fill=blue!10,draw=blue!20,rounded corners,align=center,minimum height=1.5cm,inner sep=0pt] at (7.8,4.5) {
\qquad\qquad\; $ \arraycolsep=1.5pt \begin{array}{rcc}
w_{\ms{est}}(e) = & \lambda \, \hat{p}(e) \; + & (1\!-\!\lambda) \, \hat{x}(e) \\
w(e) = & & (1\!-\!\lambda) \, x(e)
\end{array} $
};
%\node[fill=black!3,draw=blue!20,inner sep=2pt] at (7,5.4) {\includegraphics[width=1.3cm]{build/pvx-utility-anytime-simple}};
\node[fill=black!5,draw=blue!20,inner sep=2pt] at (5.6,4.5) {\includegraphics[width=1.0cm]{build/pvx-linear-discounting-simple}};
\draw[->] (7.3,3.75) -- (7.3,3.0) node [pos=0.45,fill=blue!5,align=center,inner sep=2pt] {$w$};
\draw[->] (8.2,3.75) -- (8.2,3.0) node [pos=0.45,fill=blue!5,align=center,inner sep=2pt] {$w_{\ms{est}}$};
% START LAZYSP
\node[fill=blue!10,draw=blue!20,rounded corners,minimum width=7cm,minimum height=2.5cm,anchor=north] (lazysp) at (6,3.0) {};
\node[anchor=north west] at (lazysp.north west) {\strut LazySP};
\node[fill=blue!20,draw=blue!30,rounded corners,align=center,minimum height=1cm] (dynsp) at (4.3,1.7) {DynamicSP};
\draw[->] (dynsp.south) -- (4.3,0.8) -- (7.7,0.8) -- (7.7,1.2);
\node[fill=blue!10,align=center,inner sep=2pt] at (6,0.8)
{$\pi_{\ms{candidate}}$};
\node[fill=blue!20,draw=blue!30,rounded corners,align=center,minimum height=1cm] (selector) at (7.7,1.7) {Edge Selector\\(e.g. Alternate)};
\draw[->] (selector.north) -- (7.7,2.6) -- (4.3,2.6) -- (dynsp.north);
\node[fill=blue!10,align=center,inner sep=2pt] at (6,2.6)
{$E_{\ms{changed}}$};
% END LAZYSP
% END LEMUR
% top left side
\node[fill=blue!10,draw=blue!20,rounded corners,align=center,minimum height=1.5cm,minimum width=1.8cm,inner sep=0pt] at (3.3,7.0) {};
\node[fill=white,inner sep=0pt] at (3.3,7.0) {\includegraphics[width=1.4cm]{build/c-space-simple}};
\node[font=\scriptsize] at (2.95,6.7) {$\mathcal{C}_{\mbox{\tiny free}}$};
\draw[->] (3.3,6.25) -- (3.3,5.25) node [pos=0.55,fill=blue!5,align=center,inner sep=0pt] {\strut $\mathcal{C}$};
%\node[inner sep=4pt] (cspace) at (3.3,7.0) {$\mathcal{C}$-Space};
%\draw[->] (cspace) -- (3.3,5.25);
% top right side
\node[fill=blue!10,draw=blue!20,rounded corners,align=center,minimum height=1.5cm] at (7.9,7)
{$\arraycolsep=1.5pt \begin{array}{cl}
x(\xi)\!: & \mbox{execution cost} \;(\mathcal{C}_{\ms{free}}) \\
\hat{x}(\xi)\!: & \mbox{execution cost estimate} \\
\hat{p}(\xi)\!: & \mbox{planning cost estimate}
\end{array}$};
\draw[->] (7.3,6.25) -- (7.3,5.25) node [pos=0.55,fill=blue!5,align=center,inner sep=0pt] {\strut $x$};
\draw[->] (7.9,6.25) -- (7.9,5.25) node [pos=0.55,fill=blue!5,align=center,inner sep=0pt] {\strut $\hat{x}$};
\draw[->] (8.5,6.25) -- (8.5,5.25) node [pos=0.55,fill=blue!5,align=center,inner sep=0pt] {\strut $\hat{p}$};
% left side
\draw[->] (0.5,2.0) -- (2.5,2.0);
\node[fill=white,align=center,inner sep=2pt] at (0.5,2.0)
{$q_{\ms{start}}$\\$q_{\ms{dest}}$};
% right side
\draw[->] (9.5,2.0) -- (11.3,2.0);
\node[fill=white,align=center,inner sep=2pt] at (11.5,2.0)
{$\xi$};
\only<2->{
\draw[thick,rounded corners] (dynsp.south west) rectangle (dynsp.north east);
}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Approaches for Fast Pathfinding}
\begin{tikzpicture}[font=\small]
\tikzset{>=latex} % arrow heads
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\node[fill=blue!10,draw=blue!25,rounded corners,minimum width=7cm,minimum height=1.5cm,align=center] at (4,5.95) {
Problem:\\
Dynamic Single-Pair Shortest Path (SPSP)\\
};
\begin{scope}[shift={(8.0,4.6)}]
\node[inner sep=0pt,anchor=south west] {%
\includegraphics[width=3.5cm]{figs/incbi-road-ne/singleshot/example-intro.png}};
\coordinate (s) at (1.226,0.63);
\coordinate (t) at (2.751,1.96);
\node (slab) at (1.75,0.42) {$s$};
\node (tlab) at (2.8,1.05) {$t$};
\draw[->,thick] (slab) -- (s);
\draw[->,thick] (tlab) -- (t);
\end{scope}
\only<2->{
\node at (2,2.5) {\includegraphics[width=3.5cm]{build/ibid-intro-focus-incremental}};
\node at (2,3.8) {Incremental Search};
}
\only<3->{
\node at (6,2.5) {\includegraphics[width=3.5cm]{build/ibid-intro-focus-bidirectional}};
\node at (6,3.8) {Bidirectional Search};
}
\only<4->{
\node at (10,2.5) {\includegraphics[width=3.5cm]{build/ibid-intro-focus-heuristic}};
\node at (10,3.8) {Heuristic Search};
}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Outline of Selected Algorithms}
\begin{tikzpicture}[font=\small]
\tikzset{>=latex} % arrow heads
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
%\node at (2,6.3) {\includegraphics[width=3.5cm]{build/ibid-intro-focus-incremental}};
%\node at (2,7.6) {Incremental Search};
%\node at (6,6.3) {\includegraphics[width=3.5cm]{build/ibid-intro-focus-bidirectional}};
%\node at (6,7.6) {Bidirectional Search};
%\node at (10,6.3) {\includegraphics[width=3.5cm]{build/ibid-intro-focus-heuristic}};
%\node at (10,7.6) {Heuristic Search};
\only<2->{
\node[inner sep=0pt,anchor=north] at (6,7.0) {\begin{minipage}{9.05cm}
\begin{tabular}{ccc}
\toprule
& Non-Incremental & {\only<-4>{\color{white}}Incremental} \\
\midrule
\addlinespace[0.2em]
\strut {\only<-2>{\color{white}}Unidirectional}
& {\only<-2>{\color{white}}Dijkstra's Algorithm}
& {\only<-4>{\color{white}}DynamicSWSF-FP} \\
\addlinespace[-0.2em]
\strut {\only<-3>{\color{white}}\only<12->{\color{gray}}\emph{(Heuristic)}}
& {\only<-3>{\color{white}}\only<12->{\color{gray}}\emph{A*}}
& {\only<-4>{\color{white}}\only<12->{\color{gray}}\emph{Lifelong Planning A*}} \\
\addlinespace[0.3em]
\strut {\only<-5>{\color{white}}\only<12->{\color{gray}}Bidirectional}
& {\only<-5>{\color{white}}\only<12->{\color{gray}}Bidirectional Dijkstra}
& \multirow{2}{*}{\large\only<-6>{\color{white}}\only<9-10>{\color{gray}}?} \\
\addlinespace[-0.2em]
\strut {\only<-5>{\color{white}}\only<12->{\color{gray}}\emph{(Heuristic)}}
& {\only<-5>{\color{white}}\only<12->{\color{gray}}\emph{Bidirectional A*}}
& \\
\addlinespace[0.2em]
\bottomrule
\end{tabular}
\end{minipage}};
}
\only<8->{
\node[fill=blue!5,draw=blue!15,rounded corners] at (6,4) {\begin{minipage}{10cm}Key Insights:\end{minipage}};
}
\only<9->{
\node[fill=black!5,draw=black!15,rounded corners,
anchor=north,minimum width=3.0cm,minimum height=2.5cm] (trustblock) at (2.5,3.6) {};
\node[fill=white,rounded corners,inner sep=2pt,below=0.1cm of trustblock.north]
{\includegraphics[height=1.7cm]{build/ibid-dijkstra-trust-build,winctrust}};
\node[align=center,font=\footnotesize,anchor=south] at (trustblock.south) {Trust Regions};
}
\only<10->{
\node[fill=black!5,draw=black!15,rounded corners,
anchor=north,minimum width=3.0cm,minimum height=2.5cm] (termblock) at (6.0,3.6) {};
\node[fill=white,rounded corners,inner sep=2pt,below=0.1cm of termblock.north]
{\includegraphics[height=1.5cm]{build/ibid-bidijkstra-viz-build,e}};
\node[align=center,font=\footnotesize,anchor=south] at (termblock.south) {Bidirectional\\Termination};
}
\only<11->{
\node[fill=black!5,draw=black!15,rounded corners,
anchor=north,minimum width=3.0cm,minimum height=2.5cm] (potblock) at (9.5,3.6) {};
\node[fill=white,rounded corners,inner sep=2pt,below=0.1cm of potblock.north]
{\includegraphics[height=1.5cm]{build/ibid-potentials}};
\node[align=center,font=\footnotesize,anchor=south] at (potblock.south) {Potential\\Functions};
}
\only<12>{
\draw[thick,rounded corners] (trustblock.south west) rectangle (trustblock.north east);
}
%\only<8-10>{
%\node[inner sep=0pt,anchor=north] at (6,2.3) {\begin{minipage}{10.9cm}
% {\bf 3 Key Insights:}
% \begin{itemize}
% \item {\only<8-9>{\color{gray}}
% Non-Incremental and Incremental methods both create \emph{trust regions (TRs)}}
% \item {\color{gray}
% Bidirectional termination can be rewritten w.r.t. edges between TRs}
% \item {\color{gray}
% Heuristic search is equivalent to search over a transformed graph}
% \end{itemize}
%\end{minipage}};
%}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Distance Function Approximations \& Trust Regions}
\begin{tikzpicture}[font=\small]
\tikzset{>=latex} % arrow heads
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
% top
\only<18->{
\node[fill=black!5,minimum width=5.75cm] at (3,7.5) {\strut Non-Incremental Search};
}
\only<19->{
\node[fill=black!5,minimum width=5.75cm] at (9,7.5) {\strut Incremental Search};
}
% middle
\only<1-19>{
\node[circle,align=center,fill=green!10,draw=green!50!black,inner sep=0pt,minimum width=0.5cm] (wa) at (6.0,6.8) {$w$};
\only<3> {
\node[anchor=north,fill=black!5,draw=black!15,rounded corners] at (6.0,5.5) {%
\includegraphics[width=3.5cm]{figs/incbi-road-ne/singleshot/example-dijkstraall.png}};
}
\only<2->{
\node[circle,fill=blue!10,draw=blue!20,densely dashed,inner sep=0pt,minimum width=0.6cm] (dstara) at (6.0,6.0) {\strut $d^*$};
\draw[->] (wa) -- (dstara);
}
}
\only<20->{
\node[circle,align=center,fill=green!10,draw=green!50!black,inner sep=0pt,minimum width=0.5cm] (wa) at (6.0,6.8) {$w_a$};
\node[circle,fill=blue!10,draw=blue!20,densely dashed,inner sep=0pt,minimum width=0.6cm] (dstar1) at (6.0,6.0) {\strut $d^*_a$};
\draw[->] (wa) -- (dstara);
}
\only<21->{
\node[circle,align=center,fill=green!10,draw=green!50!black,inner sep=0pt,minimum width=0.5cm] (wb) at (6.0,4.8) {$w_b$};
\node[circle,fill=blue!10,draw=blue!20,densely dashed,inner sep=0pt,minimum width=0.6cm] (dstarb) at (6.0,4.0) {\strut $d^*_b$};
\draw[->] (wb) -- (dstarb);
}
% left side
\only<4->{
\node[circle,fill=blue!10,draw=blue!20,inner sep=0pt,minimum width=0.6cm] (ld1) at (2.5,6.0) {\strut $d_1$};
}
\only<5->{
\node[align=center,fill=green!10,draw=green!50!black,rounded corners,anchor=west] (er) at (3.2,5.5) {\strut Edge\\\strut Relaxation};
\node[circle,fill=green,draw=green!50!black,inner sep=1.5pt] at (er.west) {};
}
\only<6->{
\draw[->] (wa) -- (4.0,6.8) -- (4.0,6.0);
}
\only<7->{
\node[circle,fill=blue!10,draw=blue!20,inner sep=0pt,minimum width=0.6cm] (ld2) at (2.5,5.0) {\strut $d_2$};
\draw[->] (ld1) .. controls (3.0,5.6) and (3.0,5.4) .. (ld2) node [midway,circle,fill=green,draw=green!50!black,inner sep=1.5pt] {};
}
\only<14->{
\node[fill=black!5,draw=black!15,rounded corners,align=center,
anchor=north,minimum width=4.8cm,minimum height=3cm] (lefttrustblock) at (2.5,3.6) {};
}
\only<15->{
\node[anchor=north west] at (lefttrustblock.north west) {$d \geq d^*$};
}
\only<16->{
\node[anchor=north east] at (lefttrustblock.north east) {$w \geq 0$};
}
\only<17->{
\node[anchor=south,align=center,font=\footnotesize] at (lefttrustblock.south)
{If $D$ is the minimal $d(u)$ among\\
tensioned edges $e_{uv}$, then any\\
$v$ with $d(v) \leq D$ has $d\!=\!d^*$.};
\node[fill=white,rounded corners,inner sep=2pt,below=0.1cm of lefttrustblock.north] {\includegraphics[width=2.0cm]{build/ibid-dijkstra-trust-build,wtrust}};
}
\only<8->{
\node[circle,fill=blue!10,draw=blue!20,inner sep=0pt,minimum width=0.6cm] (ld3) at (2.5,4.0) {\strut $d_3$};
\draw[->] (ld2) .. controls (3.0,4.6) and (3.0,4.4) .. (ld3) node [midway,circle,fill=green,draw=green!50!black,inner sep=1.5pt] {};
}
\only<9-12>{
\begin{scope}
\clip (1,3.5) rectangle (4,4.5);
\node[circle,fill=blue!10,draw=blue!20,inner sep=0pt,minimum width=0.6cm] (ld4) at (2.5,3.0) {\strut $d_3$};
\draw[->] (ld3) .. controls (3.0,3.6) and (3.0,3.4) .. (ld4) node [midway,circle,fill=green,draw=green!50!black,inner sep=1.5pt] {};
\end{scope}
}
\only<10-12>{
\node[circle,fill=blue!10,draw=blue!20,inner sep=0pt,minimum width=0.6cm] (ldn) at (2.5,2.5) {\strut $d_n$};
\begin{scope}
\clip (1,2.0) rectangle (4,3.0);
\node[circle,fill=blue!10,draw=blue!20,inner sep=0pt,minimum width=0.6cm] (ldnm1) at (2.5,3.5) {\strut $d_{n-1}$};
\draw[->] (ldnm1) .. controls (3.0,3.1) and (3.0,2.9) .. (ldn) node [midway,circle,fill=green,draw=green!50!black,inner sep=1.5pt] {};
\end{scope}
}
\only<11->{
\node[align=center,fill=black!5,draw=black!10,rounded corners] (ifd) at (1.0,6.0) {$d \geq d^*$};
}
\only<12>{
\node[align=center,fill=black!5,draw=black!10,rounded corners] (thend) at (1.0,2.5) {then $d_n\!=\!d^*$};
\draw[->] (ifd) -- (thend);
}
\only<22->{
\node[align=center,fill=red!30,draw=red!50!black,rounded corners] (ifd) at (1.0,4.0) {$d \ngeq d^*$};
}
\only<23->{
\fill[white,opacity=0.5] (0,0.5) rectangle (5,3.6);
}
% right side
\only<24->{
\node[circle,fill=blue!10,draw=blue!20,inner sep=0pt,minimum width=0.6cm] (rd1) at (9.5,6.0) {\strut $d_1$};
}
\only<25->
{
\node[align=center,fill=green!10,draw=green!50!black,rounded corners,anchor=east] (vc) at (8.8,5.5) {\strut Vertex\\\strut Consistency};
\node[circle,fill=green,draw=green!50!black,inner sep=1.5pt] at (vc.east) {};
}
\only<26->{
\draw[->] (wa) -- (8.0,6.8) -- (8.0,6.0);
}
\only<27->{
\node[circle,fill=blue!10,draw=blue!20,inner sep=0pt,minimum width=0.6cm] (rd2) at (9.5,5.0) {\strut $d_2$};
\draw[->] (rd1) .. controls (9.0,5.6) and (9.0,5.4) .. (rd2) node [midway,circle,fill=green,draw=green!50!black,inner sep=1.5pt] {};
}
\only<28->{
\node[circle,fill=blue!10,draw=blue!20,inner sep=0pt,minimum width=0.6cm] (rd3) at (9.5,4.0) {\strut $d_3$};
\draw[->] (rd2) .. controls (9.0,4.6) and (9.0,4.4) .. (rd3) node [midway,circle,fill=green,draw=green!50!black,inner sep=1.5pt] {};
}
\only<29->{
\node[fill=black!5,draw=black!15,rounded corners,
anchor=north,minimum width=4.8cm,minimum height=3cm] (righttrustblock) at (9.5,3.6) {};
\node[align=center,font=\footnotesize,anchor=south] at (righttrustblock.south)
{If $K$ is the minimal
$k\!=\!\min(d,\!r)$ \\
among $V_{\ms{\tiny incons}}$, then any consistent \\
$v$ with $d(v) \leq K$ has $d\!=\!d^*$.};
\node[anchor=north west] at (righttrustblock.north west) {$w > 0$};
\node[anchor=north east,color=black!15] at (righttrustblock.north east) {$d \geq d^*$};
\node[fill=white,rounded corners,inner sep=2pt,below=0.1cm of righttrustblock.north] {\includegraphics[width=2.0cm]{build/ibid-dijkstra-trust-build,winctrust}};
}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Outline of Selected Algorithms}
\begin{tikzpicture}[font=\small]
\tikzset{>=latex} % arrow heads
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\node[inner sep=0pt,anchor=north] at (6,7.0) {\begin{minipage}{9.05cm}
\begin{tabular}{ccc}
\toprule
& Non-Incremental & Incremental \\
\midrule
\addlinespace[0.2em]
\strut Unidirectional
& Dijkstra's Algorithm
& DynamicSWSF-FP \\
\addlinespace[-0.2em]
\strut {\color{gray}\emph{(Heuristic)}}
& {\color{gray}\emph{A*}}
& {\color{gray}\emph{Lifelong Planning A*}} \\
\addlinespace[0.3em]
\strut {\only<1>{\color{gray}}Bidirectional}
& {\only<1>{\color{gray}}Bidirectional Dijkstra}
& \multirow{2}{*}{\large\color{gray}?} \\
\addlinespace[-0.2em]
\strut {\color{gray}\emph{(Heuristic)}}
& {\color{gray}\emph{Bidirectional A*}}
& \\
\addlinespace[0.2em]
\bottomrule
\end{tabular}
\end{minipage}};
\node[fill=blue!5,draw=blue!15,rounded corners] at (6,4) {\begin{minipage}{10cm}Key Insights:\end{minipage}};
\node[fill=black!5,draw=black!15,rounded corners,
anchor=north,minimum width=3.0cm,minimum height=2.5cm] (trustblock) at (2.5,3.6) {};
\node[fill=white,rounded corners,inner sep=2pt,below=0.1cm of trustblock.north]
{\includegraphics[height=1.7cm]{build/ibid-dijkstra-trust-build,winctrust}};
\node[align=center,font=\footnotesize,anchor=south] at (trustblock.south) {Trust Regions};
\node[fill=black!5,draw=black!15,rounded corners,
anchor=north,minimum width=3.0cm,minimum height=2.5cm] (termblock) at (6.0,3.6) {};
\node[fill=white,rounded corners,inner sep=2pt,below=0.1cm of termblock.north]
{\includegraphics[height=1.5cm]{build/ibid-bidijkstra-viz-build,e}};
\node[align=center,font=\footnotesize,anchor=south] at (termblock.south) {Bidirectional\\Termination};
\node[fill=black!5,draw=black!15,rounded corners,
anchor=north,minimum width=3.0cm,minimum height=2.5cm] (potblock) at (9.5,3.6) {};
\node[fill=white,rounded corners,inner sep=2pt,below=0.1cm of potblock.north]
{\includegraphics[height=1.5cm]{build/ibid-potentials}};
\node[align=center,font=\footnotesize,anchor=south] at (potblock.south) {Potential\\Functions};
\only<2>{
\draw[thick,rounded corners] (termblock.south west) rectangle (termblock.north east);
}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Bidirectional Search: Termination Condition}
\begin{tikzpicture}[font=\small]
\tikzset{>=latex} % arrow heads
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\only<1>{\node at (6.0,5.8) {\includegraphics{build/ibid-bidijkstra-viz-build,a}};}
\only<2>{\node at (6.0,5.8) {\includegraphics{build/ibid-bidijkstra-viz-build,b}};}
\only<3>{\node at (6.0,5.8) {\includegraphics{build/ibid-bidijkstra-viz-build,c}};}
\only<4>{\node at (6.0,5.8) {\includegraphics{build/ibid-bidijkstra-viz-build,d}};}
\only<5>{\node at (6.0,5.8) {\includegraphics{build/ibid-bidijkstra-viz-build,e}};}
\only<6-7>{\node at (6.0,5.8) {\includegraphics{build/ibid-bidijkstra-viz-build,f}};}
\only<8->{\node at (6.0,5.8) {\includegraphics{build/ibid-bidijkstra-viz-build,g}};}
\only<1-9>{
\node[anchor=north] at (6,3.7) {\begin{minipage}{9.5cm}
\begin{algorithmic}[1]
\Procedure{BidirectionalTerminationA}{}
\State Build approximations $d_s$ and $d_t$ (alternate arbitrarily).
\State Stop on first vertex $x$ CLOSED by both sides.
\State Return path by walking $d_s$ from $x$ to $s$
and $d_t$ from $x$ to $t$.
\EndProcedure
\end{algorithmic}
\end{minipage}};
}
\only<10-12>{
\node[anchor=north] at (6,3.7) {\begin{minipage}{9.5cm}
\begin{algorithmic}[1]
\Procedure{BidirectionalTerminationB}{}
\State $p_{\ms{sofar}} \leftarrow$ best path found so far.
\State Build approximations $d_s$ and $d_t$ (alternate arbitrarily).
\State When $s$-expanding $u$, if edge $e_{uv}$ has $v$ is in CLOSED$_t$,
\State \qquad Update $p_{\ms{sofar}}$ if $d_s(u) + w(e_{uv}) + d_t(v)$ is better.
\State (Do the same in the reverse direction.)
\State Stop on first vertex $x$ CLOSED by both sides.
\State Return path $p_{\ms{sofar}}$.
\EndProcedure
\end{algorithmic}
\end{minipage}};
}
\only<13->{
\node[anchor=north] at (6,3.7) {\begin{minipage}{9.5cm}
\definecolor{greenhighlight}{rgb}{0.8, 1.0, 0.8}
\begin{algorithmic}[1]
\Procedure{BidirectionalTerminationC}{}
\State $p_{\ms{sofar}} \leftarrow$ best path found so far.
\State Build approximations $d_s$ and $d_t$ (alternate arbitrarily).
\State When $s$-expanding $u$, if edge $e_{uv}$ has $v$ is in CLOSED$_t$,
\State \qquad Update $p_{\ms{sofar}}$ if $d_s(u) + w(e_{uv}) + d_t(v)$ is better.
\State (Do the same in the reverse direction.)
\State \colorbox{greenhighlight}{Stop on when $\mbox{len}(p_{\ms{sofar}}) \leq D_s + D_t$.}
\State Return path $p_{\ms{sofar}}$.
\EndProcedure
\end{algorithmic}
\end{minipage}};
}
\only<7-9>{\node[color=red,draw,anchor=west] at (6,1.5) {\strut Unsound!};}
\only<11>{\node[color=green!50!black,draw,anchor=west] at (6,0.4) {\strut Sound};}
\only<12>{\node[color=green!50!black,draw,anchor=west] at (6,0.4) {\strut Sound, {\color{red} but overconservative.}};}
% from illustration of bidirectional termination problem!
\only<9>{
\fill[white,opacity=0.9] (2,3.6) rectangle (10,7.9);
\node[fill=blue!5,rounded corners,minimum width=6cm,minimum height=2.5cm] at (6.0,5.8) {};
\begin{scope}[shift={(3.75,5.4)}]
\node[fill=black,circle,inner sep=1.2pt] (s) at (0,0) {};
\node[fill=black,circle,inner sep=1.2pt] (a) at (1.5,0) {};
\node[fill=black,circle,inner sep=1.2pt] (b) at (3.0,0) {};
\node[fill=black,circle,inner sep=1.2pt] (t) at (4.5,0) {};
\node[fill=black,circle,inner sep=1.2pt] (c) at (2.25,0.8) {};
\draw[->] (s) -- (a) node[midway,fill=blue!5,circle,inner sep=1pt] {3};
\draw[->] (a) -- (b) node[midway,fill=blue!5,circle,inner sep=1pt] {3};
\draw[->] (b) -- (t) node[midway,fill=blue!5,circle,inner sep=1pt] {3};
\draw[->] (a) -- (c) node[midway,fill=blue!5,circle,inner sep=1pt] {2};
\draw[->] (c) -- (b) node[midway,fill=blue!5,circle,inner sep=1pt] {2};
\node[below=0.07cm of s] {$s$};
\node[below=0.07cm of a] {$a$};
\node[below=0.07cm of b] {$b$};
\node[below=0.07cm of t] {$t$};
\node[above=0.07cm of c] {$c$};
% represent closed sets
\node[fill=blue,circle,inner sep=1.2pt,below left=0.05cm of s] {};
\node[fill=blue,circle,inner sep=1.2pt,below left=0.05cm of a] {};
\node[draw=blue,circle,inner sep=1.2pt,below left=0.05cm of b] {};
\node[fill=blue,circle,inner sep=1.2pt,above left=0.05cm of c] {};
\node[fill=red,circle,inner sep=1.2pt,below right=0.05cm of t] {};
\node[fill=red,circle,inner sep=1.2pt,above right=0.05cm of c] {};
\node[draw=red,circle,inner sep=1.2pt,below right=0.05cm of a] {};
\node[fill=red,circle,inner sep=1.2pt,below right=0.05cm of b] {};
\end{scope}
}
% other example
\only<14>{
\fill[white,opacity=0.9] (2,3.6) rectangle (10,7.9);
\node[fill=blue!5,rounded corners,minimum width=6cm,minimum height=2.5cm] at (6.0,5.8) {};
\begin{scope}[shift={(3.75,5.4)}]
\node[fill=black,circle,inner sep=1.2pt] (s) at (0,0) {};
\node[fill=black,circle,inner sep=1.2pt] (a) at (1.5,0) {};
\node[fill=black,circle,inner sep=1.2pt] (b) at (3.0,0) {};
\node[fill=black,circle,inner sep=1.2pt] (t) at (4.5,0) {};
\node[fill=black,circle,inner sep=1.2pt] (c) at (1.5,0.8) {};
\node[fill=black,circle,inner sep=1.2pt] (d) at (3.0,0.8) {};
\draw[->] (s) -- (a) node[midway,fill=white,circle,inner sep=1pt] {3};
\draw[->] (a) -- (b) node[midway,fill=white,circle,inner sep=1pt] {3};
\draw[->] (b) -- (t) node[midway,fill=white,circle,inner sep=1pt] {3};
\draw[->] (a) -- (c) node[midway,fill=white,circle,inner sep=1pt] {2};
\draw[->] (d) -- (b) node[midway,fill=white,circle,inner sep=1pt] {2};
\node[below=0.07cm of s] {$s$};
\node[below=0.07cm of a] {$a$};
\node[below=0.07cm of b] {$b$};
\node[below=0.07cm of t] {$t$};
\node[above=0.07cm of c] {$c$};
\node[above=0.07cm of d] {$d$};
% represent closed sets
\node[fill=blue,circle,inner sep=1.2pt,below left=0.05cm of s] {};
\node[fill=blue,circle,inner sep=1.2pt,below left=0.05cm of a] {};
\node[draw=blue,circle,inner sep=1.2pt,below left=0.05cm of b] {};
\node[fill=blue,circle,inner sep=1.2pt,above left=0.05cm of c] {};
\node[fill=red,circle,inner sep=1.2pt,below right=0.05cm of t] {};
\node[fill=red,circle,inner sep=1.2pt,above right=0.05cm of d] {};
\node[draw=red,circle,inner sep=1.2pt,below right=0.05cm of a] {};
\node[fill=red,circle,inner sep=1.2pt,below right=0.05cm of b] {};
\end{scope}
}
%\node[inner sep=0pt,anchor=south east] at (11.9,0.1) {\scriptsize Amoebas!};
\only<8-9>{
%\begin{scope}
%\clip (0,0) rectangle (12,8);
\node[inner sep=0cm] at (6,0.35) {\begin{minipage}{11.5cm}\scriptsize
\hangindent=0.35cm \raggedright
\PaperPortrait\; Pohl.
``Bi-directional and Heuristic Search in Path Problems.''
PhD thesis, Stanford University, 1969.
\end{minipage}};
%\end{scope}
}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Bidirectional Search: Terminating with Trust Regions}
\begin{tikzpicture}[font=\small]
\tikzset{>=latex} % arrow heads
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\node[anchor=north] at (6,7.5) {\begin{minipage}{9.5cm}
\definecolor{greenhighlight}{rgb}{0.8, 1.0, 0.8}
\begin{algorithmic}[1]
\Procedure{BidirectionalTerminationC}{}
\State $p_{\ms{sofar}} \leftarrow$ best path found so far.
\State Build approximations $d_s$ and $d_t$ (alternate arbitrarily).
\State When $s$-expanding $u$, if edge $e_{uv}$ has $v$ is in CLOSED$_t$,
\State \qquad Update $p_{\ms{sofar}}$ if $d_s(u) + w(e_{uv}) + d_t(v)$ is better.
\State (Do the same in the reverse direction.)
\State Stop on when $\mbox{len}(p_{\ms{sofar}}) \leq D_s + D_t$.
\State Return path $p_{\ms{sofar}}$.
\EndProcedure
\end{algorithmic}
\end{minipage}};
\only<2>{
\draw[->,thick] (6,4.0) -- (6,3.1);
\node[fill=blue!10] at (6,2.0) {\begin{minipage}{11cm}
Define $E_{\ms{conn}}$ as the set of all edges $e_{uv}$ such that
$d_s(u) \leq D_s$ and $d_t(v) \leq D_t$,
and define $\ell_e$ s.t. $\ell_e(e_{uv}) = d_s(u) + w(e_{uv}) + d_t(v)$.
If $w \geq 0$,
$s \neq t$,
$E_{\ms{conn}}$ is non-empty,
and $e^*_{uv}$ minimizes $\ell_e$
among $E_{\ms{conn}}$ with $\ell_e(e^*_{uv}) \leq D_s + D_t$,
then $\ell_e(e^*_{uv})$ is the length of a shortest path,
and $e^*_{uv}$ lies on one such path.
\end{minipage}};
}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Outline of Selected Algorithms}
\begin{tikzpicture}[font=\small]
\tikzset{>=latex} % arrow heads
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\node[inner sep=0pt,anchor=north] at (6,7.0) {\begin{minipage}{9.05cm}
\begin{tabular}{ccc}
\toprule
& Non-Incremental & Incremental \\
\midrule
\addlinespace[0.2em]
\strut Unidirectional
& Dijkstra's Algorithm
& DynamicSWSF-FP \\
\addlinespace[-0.2em]
\strut {\color{gray}\emph{(Heuristic)}}
& {\color{gray}\emph{A*}}
& {\color{gray}\emph{Lifelong Planning A*}} \\
\addlinespace[0.3em]
\strut Bidirectional
& Bidirectional Dijkstra
& \only<1>{\multirow{2}{*}{\large\color{gray}?}}
\only<2>{IBiD} \\
\addlinespace[-0.2em]
\strut {\color{gray}\emph{(Heuristic)}}
& {\color{gray}\emph{Bidirectional A*}}
& \only<2>{\color{gray}?} \\
\addlinespace[0.2em]
\bottomrule
\end{tabular}
\end{minipage}};
\node[fill=blue!5,draw=blue!15,rounded corners] at (6,4) {\begin{minipage}{10cm}Key Insights:\end{minipage}};
\node[fill=black!5,draw=black!15,rounded corners,
anchor=north,minimum width=3.0cm,minimum height=2.5cm] (trustblock) at (2.5,3.6) {};
\node[fill=white,rounded corners,inner sep=2pt,below=0.1cm of trustblock.north]
{\includegraphics[height=1.7cm]{build/ibid-dijkstra-trust-build,winctrust}};
\node[align=center,font=\footnotesize,anchor=south] at (trustblock.south) {Trust Regions};
\node[fill=black!5,draw=black!15,rounded corners,
anchor=north,minimum width=3.0cm,minimum height=2.5cm] (termblock) at (6.0,3.6) {};
\node[fill=white,rounded corners,inner sep=2pt,below=0.1cm of termblock.north]
{\includegraphics[height=1.5cm]{build/ibid-bidijkstra-viz-build,e}};
\node[align=center,font=\footnotesize,anchor=south] at (termblock.south) {Bidirectional\\Termination};
\node[fill=black!5,draw=black!15,rounded corners,
anchor=north,minimum width=3.0cm,minimum height=2.5cm] (potblock) at (9.5,3.6) {};
\node[fill=white,rounded corners,inner sep=2pt,below=0.1cm of potblock.north]
{\includegraphics[height=1.5cm]{build/ibid-potentials}};
\node[align=center,font=\footnotesize,anchor=south] at (potblock.south) {Potential\\Functions};
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Incremental Bidirectional Dijkatra's Algorithm (IBiD)}
\begin{tikzpicture}[font=\small]
\tikzset{>=latex} % arrow heads
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\only<2->{
\node[fill=black!5,draw=black!15,rounded corners,
minimum width=3.0cm,minimum height=2.5cm] (trustblock) at (2.6,6.5) {};
\node[fill=white,rounded corners,inner sep=2pt,below=0.1cm of trustblock.north]
{\includegraphics[height=1.7cm]{build/ibid-dijkstra-trust-build,winctrust}};
\node[align=center,font=\footnotesize,anchor=south] at (trustblock.south) {Trust Regions};
\node[fill=black!5,draw=black!15,rounded corners,
minimum width=3.0cm,minimum height=2.5cm] (termblock) at (9.4,6.5) {};
\node[fill=white,rounded corners,inner sep=2pt,below=0.1cm of termblock.north]
{\includegraphics[height=1.5cm]{build/ibid-bidijkstra-viz-build,e}};
\node[align=center,font=\footnotesize,anchor=south] at (termblock.south) {Bidirectional\\Termination};
}
\node[anchor=north,inner sep=0pt] at (6,4.5) {\includegraphics[width=5.5cm]{build/ibid-bidijkstra-sep}};
\node[fill=blue!10,draw=blue!40!black,rounded corners,align=center,minimum width=3.1cm,anchor=north] at (1.6,4.5) {$Q_s$: $d_s$ Inconsistent\\Vertex Queue\\(DynamicSWSF-FP)};
\node[fill=red!10,draw=red!40!black,rounded corners,align=center,minimum width=3.1cm,anchor=north] at (10.4,4.5) {$Q_t$: $d_t$ Inconsistent\\Vertex Queue\\(DynamicSWSF-FP)};
\only<3->{
\node[fill=green!20,draw=green!40!black,rounded corners,align=center] (connblock) at (6,5.3) {$Q_c$: Connection\\Edge Queue};
\draw[->,thick] (trustblock) -- (connblock);
\draw[->,thick] (termblock) -- (connblock);
}
\only<4->{
\node[fill=black!5,draw=black!15,rounded corners] at (6,0.7) {Given $w > 0$, IBiD is {\bf complete} and {\bf optimal}.};
}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Traffic Problem Results}
\begin{tikzpicture}[font=\small]
\tikzset{>=latex} % arrow heads
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\node[fill=blue!5,draw=blue!15,rounded corners] at (3.2,6.3) {\begin{minipage}{6.1cm}
9th DIMACS Implementation Challenge
$G$: 1,524,453 vertices and 3,868,020 edges
\vspace{0.15cm}
Traffic model:
\vspace{-0.1cm}
\begin{itemize}
\setlength\itemsep{-0.05cm}
\item Independently blocked edges
\item $P_{\ms{blocked}} = 0.002$
\item Transitions: $P_{\ms{block}}: 0.0010 \mbox{ to } 0.0001$
\item 10 planning episodes
\end{itemize}
\end{minipage}};
\node[draw=black!30,inner sep=0pt] (exdijk) at (1.8,3.4) {
\includegraphics[width=2.7cm]{figs/incbi-road-ne/singleshot/example-dijkstra.png}};
\node[draw=black!30,inner sep=0pt] (exdynswsf) at (4.7,3.4) {
\includegraphics[width=2.7cm]{figs/incbi-road-ne/singleshot/example-incuni-1.png}};
\node[draw=black!30,inner sep=0pt] (exbidijk) at (1.8,1.2) {
\includegraphics[width=2.7cm]{figs/incbi-road-ne/singleshot/example-bidijkstra.png}};